Browsing by Author "Chang, Lo-Bin"

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Asymptotic error bounds for kernel-based Nystrom low-rank approximation matrices
(2013-09-01)
Chang, Lo-Bin
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Bai, Zhidong
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Huang, Su-Yun
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Hwang, Chii-Ruey
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應用數學系
Many kernel-based learning algorithms have the computational load scaled with the sample size n due to the column size of a full kernel Gram matrix K. This article considers the Nystrom low-rank approximation. It uses a reduced kernel (K) over cap, which is n x m, consisting of m columns (say columns i(1), i(2),..., i(m)) randomly drawn from K. This approximation takes the form K approximate to (K) over capU(-1)(K) over cap (T), where U is the reduced m x m matrix formed by rows, i(1),i(2),..., i(m) of (K) over cap. Often m is much smaller than the sample size n resulting in a thin rectangular reduced kernel, and it leads to learning algorithms scaled with the column size m. The quality of matrix approximations can be assessed by the closeness of their eigenvalues and eigenvectors. In this article, asymptotic error bounds on eigenvalues and eigenvectors are derived for the Nystrom low-rank approximation matrix. (C) 2013 Elsevier Inc. All rights reserved.
Empirical scaling laws and the aggregation of non-stationary data
(2013-10-15)
Chang, Lo-Bin
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Geman, Stuart
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應用數學系
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Department of Applied Mathematics
Widely cited evidence for scaling (self-similarity) of the returns of stocks and other securities is inconsistent with virtually all currently-used models for price movements. In particular, state-of-the-art models provide for ubiquitous, irregular, and often times high-frequency fluctuations in volatility ("stochastic volatility"), both intraday and across the days, weeks, and years over which data is aggregated in demonstrations of self-similarity of returns. Stochastic volatility renders these models, which are based on variants and generalizations of random walks, incompatible with self-similarity. We show here that empirical evidence for self-similarity does not actually contradict the analytic lack of self-similarity in these models. The resolution of the mismatch between models and data can be traced to a statistical consequence of aggregating large amounts of non-stationary data. (C) 2013 The Authors. Published by Elsevier B.V. All rights reserved.
Invariance in the recurrence of large returns and the validation of models of price dynamics
(2013-08-09)
Chang, Lo-Bin
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Geman, Stuart
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Hsieh, Fushing
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Hwang, Chii-Ruey
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應用數學系
Starting from a robust, nonparametric definition of large returns ("excursions"), we study the statistics of their occurrences, focusing on the recurrence process. The empirical waiting-time distribution between excursions is remarkably invariant to year, stock, and scale (return interval). This invariance is related to self-similarity of the marginal distributions of returns, but the excursion waiting-time distribution is a function of the entire return process and not just its univariate probabilities. Generalized autoregressive conditional heteroskedasticity (GARCH) models, market-time transformations based on volume or trades, and generalized (Levy) random-walk models all fail to fit the statistical structure of excursions.